Proposition: Sharp instruments and a sufficient dephasing condition

Let the fine measurement be a PVM $\{P_z\}$, and the coarse measurement a PVM $\{E_w\}$ with Lüders maps ${\mathcal{I}}_w(X)=E_{w}X{E}_w$. Assume $T^{\ast }(E_w)={\Pi }_w$ for every $w$. Then

The full instruments $\mathcal{F},\mathcal{G}$ agree if and only if

In particular, for $\mathcal{D}(X)={\sum }_z{P}_{z}X{P}_z$, the additional condition

is sufficient.

Proof. Choose Kraus operators $K_a$ for $T$. For $v\in \mathrm{ran}{P}_z$, pullback compatibility gives

Thus $E_w{K}_a{P}_z=1_{\{q(z)=w\}}{K}_a{P}_z$. Summing the Kraus formula proves eq:sharp-block-intertwining.

Consequently each coarse block is

The fine block retains only its $z=z^{\prime }$ terms. Condition (eq:intra-outcome-condition) is sufficient. If instruments agree for all states, linear spanning extends the identity to every $X$. Substitute a single off-diagonal block $X=P_{z}Y{P}_{z^{\prime }}$ with equal coarse label to obtain necessity.

Finally, $T=T\circ \mathcal{D}$ kills every fine off-diagonal block.